11 THE SPACES L1 AND L2 163 We remark that, given /e JS?i(X, /z), it is not always possible to find a function/' equivalent to /and a sequence of functions/, e «?TR(X) such that the sequence (/„(*)) converges to/'O) for allxeX (Problem 3). We have already defined (13.10) the vector space -S?£(X, /*) of complex- valued integrable functions. Likewise, we denote by <£ £(X, ju) (or <£ cOu), or JS?£) the set of c0ra/?/ex-valued /i-measurable functions such that \f\2 is ji-integrable, and we define N^/) and N2(/) by the formula (13.11.1) for any complex-valued function/ Since a complex-valued function is measurable if and only if ^?/and .//are measurable, and since it follows that #S(X,Ai) = JS? J(X,M) 0 fcS?2 All the properties of J$?£(X, ju) proved above extend immediately to JSf£(X,/i) Q? = l,2), and the Banach space JS? J(X, //) (also denoted by ^ c(j") or «^c) ^ defined exactly as in the real case. Furthermore, we have (13.11.7) Iff, g belong to -SfJ(X, /*) (resp. JS?£(X, ^)) then their product f g belongs to 4fi(X, ji) (resp. Jif i(X, ju)). T/ze space Lj(X, ^) (resp. Lj(X, ji)) w « separable Hilbert space with respect to the Hermitian form (13.11.7.1) (/, g) h+(/|0) = tfWtfddrtx), and the corresponding norm is N2(/). The product/^ is measurable (13.9.8.1), hence the first assertion follows from (13.9.13) and (13.11.2.2). The second follows from the first together with the Fischer-Riesz theorem (13.11.4) and (13.11.6). PROBLEMS 1. Let A be Lebesgue measure on I = [0,1 [. For each integer « = 2"+ &(()<;/:< 2"), let/n be the function which is equal to 1 on the interval [k • 2~h, (k+ 1) -2~*[, and zero elsewhere. Show that the sequence (/„) converges to 0 in mean and in square mean, but that the sequence (/„(*)) does not converge for any x e I,nough to observe that for any # e ^ R(X; K) we have